We present the results of sixteen Swift-triggered GRB follow-up observations taken with the VERITAS telescope array from January, 2007 to June, 2009. The median energy threshold and response time of these observations was 260 GeV and 320 s, respectively. Observations had an average duration of 90 minutes. Each burst is analyzed independently in two modes: over the whole duration of the observations and again over a shorter time scale determined by the maximum VERITAS sensitivity to a burst with a t-1.5 time profile. This temporal model is characteristic of GRB afterglows with high-energy, long-lived emission that have been detected by the Large Area Telescope (LAT) on-board the Fermi satellite. No significant VHE gamma-ray emission was detected and upper limits above the VERITAS threshold energy are calculated. The VERITAS upper limits are corrected for gamma-ray extinction by the extragalactic background light (EBL) and interpreted in the context of the keV emission detected by Swift. For some bursts the VHE emission must have less power than the keV emission, placing constraints on inverse Compton models of VHE emission.